BSTJ 51: 9. November 1972: Stability Considerations in Nonlinear Feedback Structures as Applied to Active Networks. (Baumwolspiner, M.)

Copyright © 1972 American Telephone and Telegraph Company

The Bell System Technical Journal

Vol. 51, No. 9, November, 1972

Printed in U.S.A.

Stability Considerations in Nonlinear

Feedback Structures as Applied to

Active Networks

By M. BAUMWOLSPINER

(Manuscript received April 24, 1972)

Active filters have recently acquired widespread use in the realization of
frequency-selective networks. Unlike their passive counterparts, active
filters have the potential of oscillating.

Furthermore, it has been observed that the onset of oscillations in biquad
active filters is dependent upon signal level. This led to the recognition
that nonlinear stability theory would be necessary to comprehend this
behavior.

This paper develops a technique to analyze the stability of networks
containing linear and nonlinear elements interconnected in multifeedback
structures. This is accomplished by extending the concept of the "Describing
Function" to include networks containing nonlinearities with frequency-
dependent linear feedback. The technique is then applied to explain
qualitatively and quantitatively nonlinear effects in op-amps and their
relation to the stability of frequency-selective networks containing them
{e.g., the Multiple Amplifier Biquad, MAB, and the Single Amplifier
Biquad, SAB). The technique is also applied to explain frequency shifts in
amplitude-limited oscillators. The most valuable result of this analysis is
the discovery of nonli?iear feedback circuits which circumvent the conditional
stability of high-frequency biquads. This has allowed us to obtain Q's of
50 at 100 kHz in a MAB employing 709 op amps. Similarly, a MAB
employing 702 op amps was made to operate at 2 MHz with a Q of 10.

I. INTRODUCTION

In this paper, we shall deal primarily with a frequency-domain
approach of analyzing networks containing linear and nonlinear elements
interconnected as multifeedback structures. Particular applications
will include the Single Amplifier Biquad 1 (SAB), the Multiple Amplifier
Biquad 2 (MAB), and amplitude stabilized oscillators.

2029

2030

THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

y m = M, SINM+ifr,) +

/ Kit CIM/n,-til 1 j

i

E SINtot

/
/

/

i ,

r^

NONLINEAR

NETWORK

N

LINEAR

NETWORK

H<jw)

r-

1

Fig. 1 — A typical system with separable networks.

Our interest in this subject is motivated by conditional stability
problems in biquads designed to operate above certain critical fre-
quencies. Above these frequencies, these filters exhibit oscillatory modes
when certain excitations are or were present. In the past, these
conditional stability problems have been attributed to slew rate limiting
in the operational amplifiers. However, we shall demonstrate later that
although slew rate limiting may affect the stability, it is neither a
necessary nor sufficient condition for conditional stability to occur
in the MAB circuit.

We shall draw heavily on the Describing Function (DF) and the
Dual Input Describing Function (DIDF) Techniques. 3-8 These tech-
niques have the virtue of imparting a conceptual understanding of the
problem. Techniques based on the Liapunov stability criterion such as
the Popov criteria and circle criteria have been looked into and seem to
offer overly restrictive sufficient conditions for stability in most of our
reasonably high Q applications. In addition, they fail to extend easily
to multiple nonlinear loops.

To facilitate the application of these techniques to the filter condi-
tional stability problem, several interesting results concerning opera-
tional amplifiers will first be derived. Earlier nonlinear op-amp models
have represented an op-amp by saturating elements intermingled with
frequency-dependent networks in a feedforward configuration. It will
be seen that this is insufficient to predict insertion phase measurements
of an op-amp in its nonlinear region. However, the usual position of
compensation elements in an op-amp is in a feedback path around the
gain stages yielding higher effective (Miller effect) capacitances. As
a consequence, the linear and nonlinear elements become interconnected
in feedback structures which impart varying phase characteristics as a
function of amplitude. This becomes highly significant when considering
active resonant circuits. Most important, when high Q circuits are
realized, any additional phase lag around a loop may be sufficient to
increase the pole Q to the point of oscillation. Similar phase shifts may

STABILITY IN NETWORKS

2031

occur in oscillators in which nonlinearities are purposely introduced to
stabilize their amplitude. In oscillators these effects are manifested
by a discrepancy between the linearly-computed and actual frequency
of oscillation.

Finally, we will present circuits which produce either phase lag or
lead as a function of amplitude. These circuits may be useful for non-
linear compensation of the aforementioned problems.

II. THE DESCRIBING FUNCTION TECHNIQUE

The describing function method is an outgrowth of the Harmonic
Balance technique used by Krylov and Bogoliubov, 6 in nonlinear
mechanics. The method reduces a nonlinear differential equation into
a linear relationship by assuming a sinusoidal solution. The method is
most useful when the system contains sufficient lowpass filtering to
allow higher harmonics to be neglected. However, if the describing
function technique is inadequate due to its neglect of higher order
harmonics, the DIDF may help in solving the problem.

2.1 Input-Output Concept oj A Nonlinear Element

We begin our analysis by reviewing some basic concepts of the
DF technique. Consider the system of Fig. 1, where the linear and
nonlinear parts are assumed separable. N is described in terms of its
effect on a sinusoidal waveform. In particular, the describing function
is defined as

t-)f - (w\ A Fundamental of m M t

~ g{ > ~ Fundamental of e = ~E Z 4>l ' (1)

In general, the DF will be a complex quantity. However, if AT is a
single-valued nonlinear function, its input-output characteristic will
not enclose any area (see Fig. 2) and its DF will always be real as
shown in Appendix A.

m (OUTPUT)

m (OUTPUT)

e (INPUT)

INPUT)

Fig. 2— (a) A single-valued nonlinearity (real DF). (b) A nonlinearity having a
hysteresis loop that encloses a given area (complex DF).

2032 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

On the other hand, the DIDF is denned for a two sine wave input,
with one wave at a multiple frequency of the first, i.e.,

e = E cos (at + 4>) -\- E y cos mat.

The DIDF is the ratio of the desired frequency component in the output
over the same frequency component at the input. The DIDF has been
worked out by West, et al., 7 for a saturating type of nonlinearity
operating on a sinusoidal input in the presence of the third harmonic.
This function can be used in determining the required perturbation
of the DF when the third harmonic is not sufficiently low for the DF
to be directly applicable.

With the above ideas applied to Fig. 1, we may extend linear stability
criteria to obtain the stable and unstable limit points for this case.
Assuming that there is sufficient lowpass filtering following N in Fig. 1,
the input-output relationship of the system is:

vo _ q{E)H(ju) (2)

v< " 1 + g{E)H(ju)

HOoO — (3)

' +H(ju)

g(E)

We now apply the Nyquist criterion, but instead of taking the critical
point as —1 and incorporating g(E) into ff(jw), we take — [l/g(E)] as
the critical point.

Figure 3c shows the conventional Nyquist plot and the locus of
- [I ME)) with E as a parameter for the system the block diagram of
which is given in Fig. 3a. The describing function of the nonlinearity,
as shown in Fig. 3b depicts the variation in amplitude (the phase shift
being zero) of the fundamental of sin ut operated on by a dead-zone
nonlinearity. We note that if E is small, the critical point is not encircled,
and the system will be stable. As E is increased, we reach point A, and
the system becomes unstable. As a consequence, E increases till we
reach point B where the system enters into a stable limit cycle. The
limit cycle is stable since if E increases, the system becomes stable
causing E to decrease. On the other hand, if E decreases, the system
instability is such that E will increase. The intersection point B defines
the amplitude (E) and frequency (w) of oscillation. It is to be noted
that in the above case the frequency of oscillation occurs at the inter-
section of H(ju) with the real axis since N introduces no phase shift
in this case. However, when N introduces phase shift, this will not be

STABILITY IN NETWORKS
e = E SINojt

(a)

0.6-

'g(E) =

(b)

i m H(i<">

-1/g(E))

E INCREASING

-»-E

2033

*-R e H(jco)

Fig. 3 — (a) System with dead-zone relay, (b) DF of dead-zone relay, (c) Nyquist plot.

the case. Furthermore, even when .A/ has a real DF and the second
harmonic is not sufficiently filtered out, the DIDF predicts a phase
shift of the fundamental through N.

2.2 Input-Output Concept of A Feedback Structure with Nonlinear
Elements in it

With the above as background, we proceed further with the DF
concept by determining the sinusoidal input-output relationship for a
feedback system with a nonlinearity. We will do this by considering the
specific example of Fig. 4, which resembles an op-amp with unity
feedback. As before, we assume sufficient lowpass filtering to eliminate
the effect of the second harmonic.

2034 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

^'=|E'| SIN(o)|t +£')

e = | ElSINfcjjt+Zt) /
V,-|V,|8IN( M| t+A|) / ; /V -|V |SIN(ft>it +^ I

! / H(S) I N /

*0

Aab

(s+a) (s+b)

X

Fig. 4a — System with saturating nonlinearity.

It is clear from Fig. 4a that

y _ v = E or V. = E + V .

(4)

In addition, we know the relationship between E and V from the
describing function of Fig. 4b and the linear transfer function H(S).
Therefore, we can obtain the relationship between V,- and V . This is
done most appropriately by considering the phasor diagram of Fig. 5b.
First, we obtain the transfer characteristics of the linear block,
H(ju) as shown in Fig. 5a. Incidentally, this is the Nyquist plot for the
linear region of operation. Assuming momentarily that we hold w = wi ,
H(ju) and g(E'), determine the necessary magnitude and phase of E
to give a particular V . From eq. 4, we then find the corresponding
Vi by taking the vector sum as is shown for several V Q in Fig. 5b.
Observe that the boundary between the linear and nonlinear region is
(V< 2, f y<. 2) , E m ). In the linear region ZV — £V\ is independent of
amplitude, while in the nonlinear region ZV - ^ V t is a function of
amplitude as shown in Fig. 5c. As a result of H(ju), there will be a
different curve of this type for each frequency considered. These curves
together with similar ones for | V 1/| 7, | define the DF for the system

■ g(E')=1

g IE'HHaRC S | N jjjj+^ ('-JfVI

1/2

^{E')=0

Fig. 4b — Describing function of saturating nonlinearity.

STABILITY IN NETWORKS
,l m H(jc

A

2035

Fig. 5a — Nyquist plot of H(ju).

LINE
REG

AR m

NONLINEAR/
REGION /

DN

<ti=<l) |

Vi d)
\

\

\
\
\

\

Vj (2)

\

\
\
\

v .(3)
1
t
1^

(£(2|

v .(4) /
/g(3> \.

V (1)

E(D

V (2)

V (3)

V (4)

Fig. 5b — Vector diagram for determining V,-.

\%

0= BOUNDARY BETWEEN

LINEAR AND NONLINEAR
REGIONS

Fig. 5c — Phase shift vs input signal.

of Fig. 4a. This describing function differs slightly from those considered
thus far in that it is a function of both amplitude and frequency.

A most important result of the input-output concept developed in
this section is the capability of being able to analyze a rather complicated
nonlinear network by breaking it up into its constituents. Each one is
analyzed individually and then combined as shown in Fig. 6. After

2036 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

|E t |SIN(<uit+^E 1

9i(E|. <u,)

|E 2 |SIN(ai,t+^E2>

g 2 (E 2 . oil)

|E 3 |SIN(a.,t + ^3l

G

O «

-1 |E 3 |SIN(oi 1 T + /E 3 )

g 3 (E,,w 1 ) = g,(E,,wi) g 2 [E,g, (Elw^.cdi]

Fig. 6 — Interconnection of DF's

reducing the network we apply a variation of the analysis employed
for the example of Fig. 3. The difference lies in the fact that the DF is
now a function of frequency and amplitude. As a consequence, we
obtain many DF's, each for a particular frequency, of which the negative
reciprocal must not intersect the Nyquist plot at that frequency for
stable operation. Usually, as we will see, the critical frequencies corre-
spond to those close to the critical point in the Nyquist plane.

A network which is well suited to this analysis technique is shown in
Fig. 7. This is a typical Multiple Amplifier Biquad Filter using mono-
lithic op-amps. Here, it is clear that if we employ the op-amp model
shown in Fig. 7b, we have a multifeedback structure containing several
nonlinearities. Deriving the DF of each closed loop op-amp and com-
bining results, we can determine if the phase lag around the loop at
certain amplitudes is sufficient to cause the biquad to oscillate. This
is the subject of the next section.

1 — VW^'n

R 4

4

OUT

- Aab

-I 1

y\

( s+a) Is+b)

J

(a) (b)

Fig. 7 — (a) MAB circuit, (b) A possible model for the op-amp.

STABILITY IN NETWORKS

2037

p/a =25

a = 2-8-10 3 RAD/SEC
A. = 10 3

A 2 = 10
a - 3.16 VOLTS = b
c = 15 VOLTS

-A 2 /3
S+/3

OUT

Fig. 8 — Fairchild /*A709 op-amp model.

III. APPLICATIONS

In this section we shall apply the techniques developed earlier to
op-amps, active filters, and oscillators. Since active filters (e.g., MAB
and SAB circuits) usually employ monolithic op-amps, we shall develop
first an accurate model of the linear and nonlinear aspects of the op-amp.
The op-amp model will generally depend on the type (e.g., 709, etc.)
and compensation used. However, we shall demonstrate how to arrive
at the model and present typical circuits.

3.1 Operatiofial Amplifiers

3.1.1 Open Loop Characteristics of Op-Amps

We have already presented one possible model of a typical op-amp
in Fig. 7b. It is possible to modify the model by placing an amplitude
limiter at the input or splitting the linear transfer function by inserting
an amplitude limiter. An excellent model, as viewed from pulse and
slew-rate measurements on a Fairchild /iA 709 op-amp is shown in
Fig. 8. The DF for this model does not predict any extra phase shift
as a function of amplitude. This follows from Appendix A, where it is
shown that this type nonlinearity (no hysteresis) introduces no addi-
tional phase shift. However, if we consider the DIDF, we may indeed
get extra phase lag as a function of amplitude. This results from the
presence of harmonics, generated by the first nonlinearity, at the input
of the second nonlinearity. The amount of phase lag may be estimated

Fig. 9a — Symmetrical sat urator.

2038

THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

*£&***

B N« o.

-fl

b=0.6

b=1.0

b=2.0

,a=10.0

,a=8.0
,' ,a=4.0

/ ^=2.0

! / / a=0.5

b=4.0

b=8.0

Im (-

DIDF

Re

DIDF

a = AMPLITUDE OF

FUNDAMENTAL

b = AMPLITUDE OF

THIRD HARMONIC

RADIAL LINES ARE ALL
10° APART AND
REPRESENT LINES
OF CONSTANT DIDF
PHASE; ANNULAR LINES
ARE LINES OF CONSTANT
DIDF AMPLITUDE.

"o = "i

-1<Vj < 1

I'd— 1

*i<-1

V Q = 1

*l>1

Fig. 9b — Input-output relationships for fundamental in presence of third harmonic
for saturating nonlinearity.

STABILITY IN NETWORKS

2039

by using the set of DIDF curves for a symmetrical amplitude limiter 7
shown in Fig. 9. The curves for each value of a represent the DIDF as a
function of the phase shift between the fundamental and third harmonic
at the input of the saturator.

From these figures, it can be shown that a maximum fundamental
phase lag of 11.1 degrees may be obtained when the op-amp of Fig. 8
is heavily overdriven (i.e., a square wave input to the second saturator).
Yet, when the phase characteristics of a 709 op-amp were measured
in the lab, no such effect was observed. The curves obtained are shown
in Fig. 10. These curves display phase lead as a function of amplitude
instead of phase lag. This result is extremely important, for it verifies the
stability inherent in biquads at low frequencies as we shall see later.
In summary, a better model of the op-amp is needed.

To determine this model, we first consider a simple circuit depicted
in Fig. 11. This circuit is a transistorized amplifier which provides
output limiting of the signal. Its applications include FM limiters and

2.7KO

M A709 C

n

100pf

82 pf
Q^>J HP MOD. 651 B

40° 50" 60° 70° 80° 90° 110° 120° 130° 140° 150° 160° 170°

< OUTPUT - <J INPUT

Fig. 10 — Measured phase characteristic of an open loop 709 op-amp.

2040 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

Fig. 11a — Transistor amplifier with output limiting.

gm V in = I

Fig. lib— Model of Fig. 11a assuming transistor is in linear region.

♦ 'd<Vo>

Fig. lie — Characteristics of diode pair.

amplitude-stabilized oscillators. Defining Z as

7 Rl

SCJIl + 1

it follows from the circuit that:

/ = Vo/Z + I d (V )

(5)
(6)

STABILITY IN NETWORKS

2041

'd<V >

s *r^ ii

z

' d 1

i

) ,

r

(

V o

Fig. 12a— Block diagram of Fig. lib.

or

V = Z[I - I d (V )l

(7)

This equation is represented by the block diagram of Fig. 12a. Knowing
the relationship between 7 X and l d , we proceed to draw the phasor
diagram of Fig. 12b. Initially at low levels, I d is small compared to U
because the diodes are cut off. However, if /, is just short of turning
the diodes on, a slight increase in /j will cause a large increase in I d .
These two situations are represented by {I d l) , I[ v ) and {I d 2) , I[ 2) )
respectively, in Fig. 12b. Taking their vector sum, we obtain J (1) and
I 2) . Interestingly we note that the phase difference between / and I d
and hence V iB and V„ has decreased. By taking the small signal ease as
a reference, we conclude that the closed loop network exhibits phase
lead with increasing amplitude.

A significant conclusion to be drawn from this example is that although
the simple circuit appeared to contain just a clipper (which would
indicate no amplitude-dependent phase shift) the feedback present
alters the situation sufficiently to predict phase shift as a function of
amplitude.

If the same form of analysis is carried out for the internal circuitry

< PHASE LAG OF V„ OR l d
WITH RESPECT TO I

r/2--

Fig. 12b — Input/output relationship.

2042 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

SLOPE = -m= -g m R 3

^G>

1 V a /

V.

^ E

SC,^

J

E °\

■*- 0.7 VOLT

SC,(R,+R 2 +R3)+1

(a)

v ^0

R 3 SC, (R,+ R 2 ) + 1

M R 2

SC,

|*-v SAT

(b) x SLOPE k

Fig. 13 — 709 op-amp input compensation circuity model, (a) Cutoff region, (b)
Saturation region.

of the 709 op-amp, similar results are obtained. This has been done in
Appendix B with particular attention to the circuitry involved in the
input and output compensation. Most often, compensation in an op-amp
is obtained by making use of the Miller effect to obtain low-frequency
breakpoints with relatively small capacitors. Both the input and output
compensation in a 709 op-amp take advantage of this effect. As a
result, the transistor nonlinearities have RC feedback around them.
The equivalent circuits of the input compensation circuitry in the
cutoff and saturation regions, as derived in Appendix B, are shown in
Fig. 13. In this figure we have left out some linear transfer function
blocks at the input and output side, since these have no effect on the

.211/2

FOR A >

x o

|g(A)U

9m R 3 9 m R 3

g(A)= 5 y—

--'(*)**

[-f

9 m R 3

9m R 3
2

E o -0.7 VOLTS = X

Fig. 14a — DF for nonlinear element in cutoff region model.

STABILITY IN NETWORKS

2043

nonlinear characteristics. The following analysis will show that both
of the circuits in Fig. 13 provide phase lead as amplitude increases.
Figure 14a shows the DF of the nonlinear element 8 in the cutoff
region model under the assumption that the quiescent point is at
E = E . Figure 14b depicts the phase relationships of the linear element
in that model. With this information and Fig. 13a, we construct the
phasor diagram of Fig. 14c. Specifically, since the nonlinear element
is single valued and therefore contributes no amplitude sensitive phase
shift, — V will lead E according to the linear element and the inversion
due to the nonlinear element. Entering the nonlinear region of the
model, the error voltage E has to increase at a faster rate than V to
overcome the attenuation effect of the clipping element. From the
relationship

Vi = (-V ) +E

(S)

4V„-SV a

C,(R, + R 2 +R 3 )

Fig. 14b — Phase characteristic for linear element, in cutoff region model.

LINEAR NONLINEAR
REGION | REGION

+Vq (2) Vo (1)

_v HI 1 -V (2 >

Fig. 14c — Phasor diagram for the closed loop in cutoff region model.

2044

THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

g(A)

K/a

for A>X C

-*-!-* {- ~'*i •* ^"J

Fig. 14d — DF for nonlinear element in saturation region model.

we determine the third vector V, . As a result, the angles between
(—Vo l) , V\ l} ) and (-V l 2) , V™) do change with increasing drive.
However, at high frequencies where E and — V are initially close to
being in phase, the increase of phase lead of F„ relative to V t is minimal.
The frequency breakpoint below which the nonlinear phase lead effect
may be expected is typically,

/

2w(R 1 + R 2 + B,)C,

50 kHz.

(9)

On the other hand, in the saturation region, the model of Fig. 13b
applies. Here the DF for the nonlinear element is given in Fig. 14d under
the assumption that the quiescent point is at V = V 00 . Furthermore,
we have approximated the exponential nonlincarity of the diode in a
piece-wise linear manner. It follows directly that the phase relationship
of E with respect to V is given by the phaso characteristic of the lag
network following the nonlinear element in Figure 13. Drawing the
phasor relationships for the closed loop at a fixed frequency where the
lag network has singificant phase lag yields Fig. 14f. Again, the phasor
diagram reveals that the phase lead of V with respect to V,- increases
as amplitude increases. The frequencies over which this phase lead

XE-*V

90'

Fig. He — Phase characteristics for the linear element in saturation region model.

STABILITY IN NETWORKS

2045

LINEAR REGION

NON-LINEAR REGION

Fig. 14f — Phasor diagram for the closed loop in the saturation region model.

effect will be present is approximately the range of frequencies where
the linear network provides phase lag. From Fig. 14e, this occurs for
frequencies below:

/

2ttC 1 (^ 2 4- R t )

100 kHz.

(10)

Above this frequency range, E ll) and E i2) will have approximately the
same orientation as V"' and Vq 2> , forcing 7- 11 and V l { 2) to maintain
their orientation roughly independent of amplitude.

The above analysis also applies to the output compensation circuitry
provided that it is properly interpreted. The output compensation cir-
cuitry for a 709 op-amp and its nonlinear model are shown in Fig. 15.
Here, unlike the input compensation, the RC network is fed back from
an emitter follower circuit which simplifies the model and increases the

«v out

Fig. 15a — Output compensation of 709 op-amp.

2046 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

„-SLOPE =-m

R P /R 5

H i ,s ' = ^ r, . . : r p = R 4ll R

H 2 (s) -

SC,R p + 1

S + 1/C, R 4

S + 1/CtRp

+

-

"*\— e

1

Vout

K

R 5

H 2

Fig. lob — Nonlinear model of 709 op-amp.

breakpoint frequencies. In addition, the RC network is fed back to a
lower impedance termination which also has the effect of increasing
the breakpoint frequencies. In a typical 709 op-amp, this breakpoint
frequency occurs at (the equivalent of eq. 9 with R 2 and R* being zero)

^2^A" L5X1 ° 6HZ -
This frequency is significantly higher than those resulting from the
input compensation (see eqs. 9 and 10). As a result, the output compen-
sation will be quite irrelevant to the conditional stability of biquad
filters in the to 100 kHz range. Above 100 kHz, the amplifier will no
longer be used in a biquad filter, since the open loop gain is less than
40 dB, which is insufficient for most precision applications.

We now have an op-amp model consistent with the experimental
data of Fig. 10. Having discussed these open loop characteristics of an
op-amp, we shall next use these concepts to derive the phase properties
associated with closed-loop operational amplifiers.

3.1.2 Closed-Loop Characteristics of Op-Amps

We shall confine ourselves, in this section, to gain inverters, inte-
grators, and leaky integrators. The circuit of Fig. 16a is the general case
which includes all the above-mentioned configurations. The model of
the closed loop op-amp shown in Fig. 16c is derived as follows. Assuming
that the op-amp has infinite input impedance and letting

e in = 0,
the voltage developed across the input of the op-amp will be
R, Sfl.C + g./fl<

e =

e„„t =

Ri +

fe.»*]

SRiCt + [Ri/R„ + 1]

(11)

STABILITY IN NETWORKS

2047

B, n =|V,'|COSUt+*l)

o

AMr

ti-

t=R,C, a = R,/R

'1^1

1' "q

e out =- |V | C0S(a»t + « )

o

v'

Fig. 16a — Closed loop op-amp.

ST + a

57 + 11*1.)

&

OP-AMP
MODEL

Fig. 16b— Model for Fig. 16a.

Fig. 16c— Model for Fig. 16a.

N 2 (s)

!S e t

sr+a

OP-AMP

MODEL

- V c

ST+(1+,a)

Likewise, letting

e mll = 0,
the input voltage of the op-amp will be

e =

SC,

B.

be; 11 *-]

e in =

fl, +

SR& + [/2,/fl, + 1]

(12)

The block diagram of the closed-loop op-amp, obtained through the
superposition of eqs. 11 and 12, is shown in Fig. 16b. By the reduction
method, we move the feedback network past the summing node and
make the proper correction to the input network. This leaves us with
the model of Fig. 16c. A significant advantage of this model is that
the ideal transfer function has been separately realized by the network

2048

THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

between 7 J and 7, . Consequently, the feedback network following
the ideal transfer function network gives a direct measure of the error
introduced by the nonideal characteristics of the op-amp.

For the model of the op-amp itself, we may either use the networks
of Fig. 13 in a cascade combination, or we may directly employ the
DF curves shown in Fig. 10. The latter method is to be preferred since
it is based on actual measurements and requires a minimum of computa-
tion. Fig. 17 is the phasor diagram for an inverter (i.e., Fig. 16 with
t = and a = 1) at / = 5 kHz. In the linear region, we initially obtain
a very small error voltage E and consequently 7, and V are almost
equal. This error voltage, although small, has a distinct phase lead of
approximately 70 degrees with respect to the output voltage V .
This phase shift is introcuded by the frequency characteristics of the
op-amp and the network N 2 (s) in Fig. 16c. However, for the unity gain
inverter, N 2 (j<t>) is a real quantity and, therefore, will not add additional
phase. As we enter the nonlinear region, the magnitude and phase of
the error voltage, E change as dictated by the describing function (DF)
of the op-amp. The phase changes according to Fig. 10 and the amplitude
according to the real DF of a saturation type of nonlinearity shown in
Fig. 4b. Although the op-amp does not behave exactly as a saturation
nonlinearity; the magnitude of the DF for the op-amp is well approxi-
mated by this DF, unlike the phase shift of the DF which for the satura-
tion nonlinearity is zero.

Having obtained the phasor diagram of Figure 17, we note that
increasing the input 7, predicts an increasing phase lag of the output
— 7 relative to the input 7, . This effect has been verified experimentally
as shown by the curves in Fig. 18. At 27 kHz where the op-amp has

•> ,0 U n)

Fig. 17 — Phasor diagram for unity gain inverter (using a 709 at 5kHz).

STABILITY IN NETWORKS

2049

208

204

200

CD 196

3?

188 -

184 -

176

FREQ=» 27 kHz

UNITY GAIN INVERTER

(COMP. LIGHT)

180 - O.

FREQ=*5 kHz

UNITY GAIN INVERTER

(COMP. LIGHT)

9 10 11

E, n (RMS) VOLTS

12 13 14

Fig. 18 — Measured phase characteristics of a unity gain inverter employing a 709
op-amp.

reduced gain, the effective saturation nonlinearity comes in earlier
producing the phase lag at a lower input level.

For the integrator, the same analysis applies but with the critical
difference that N a (ju) in Fig. 16c is no longer a real quantity as in the
case of the inverter. The phasor diagram for this case is shown in
Fig. 19. We have chosen an integrator with a = and t, the reciprocal
of angular frequency, 2tt-5 kHz. As a result, N 2 (ja) will provide a
45-degree lead at 5 kHz, which will have the effect of moving the phase
of the error voltage, E, 45 degrees closer to the vector V than in the

/

S*s

25°

Vj'" / Vj » 2 > \J^---v <

V (11/

LINEAR
REGION'

3) -^, /28°

NONLINEAR
REGION

Fig. 19— Phasor diagram for integrator at 5kHz,r = l/2:r5kHz, a = 0.

2050

THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

case of the inverter. However, when the system is overdriven, E swings
around to the lagging side of V , which consequently causes V to lead
Vi . As a result, in the integrator at this frequency, increasing the drive
level produces a phase lead effect. This has been verified experimentally
and the results are shown in Fig. 20.

3.2 Active Filters

3.2.1 The Multiple Amplifier Biquad Filter

The above discussion has provided a basic feel for what happens when
the op-amp is employed in a simple feedback loop. We shall next discuss
the performance of networks which use these inverters or integrators
as building blocks.

The MAB 2 circuit was already shown in Fig. 7a in its bandpass
configuration. The configuration generates a pair of poles with arbi-
trary Q and frequency location. To generate a pair of zeroes, outputs
taken at different points are summed in a separate summing amplifier.
The zeroes can also be generated by feeding the input to more than
one op-amp as is done in the Multiple Input Biquad Filter (MIB).
However, their basic operational frequency limitation manifests itself
by a conditional stability problem in its pole forming loop. This problem
is more acute when high Q's are desired, for then the resonator loop
operates very close to the critical point of its Nyquist plot. This is
shown in Fig. 21, where it is seen that the phase shift around the loop
at the pole frequency gets very close to 180 degrees as the Q is increased.

As a consequence, a small amount of additional phase lag in any of the
amplifier blocks (i.e., the inverter and the leaky and nonleaky inte-
grators) making up the main loop can make the system unstable.

With the information already acquired, as to the characteristics of

8

6

§

4

?

c:

LU

II

UJ

-2

\J

-4

-6

S FREQ.=>5 kHz. LEAKY
INTEGRATOR. LIGHT COMPENSATION
I I I I I I I

,FREQ.=>27 kHz, LEAKY
/ INTEGRATOR. LIGHT
* COMPENSATION

12 3 4 5 6 7 8

E in (RMS)

9 10 11 12 13 14

Fig. 20 — Measured phase characteristics for integrator employing a 709 op-amp
(the gain is unity).

STABILITY IN NETWORKS

2051

A ImGljto)

(o=Q K 2

Fig. 21 — Nyquist plot for main loop in MAB.

the individual amplifier blocks, the task of determining whether a
given filter will be stable can be investigated. By adding up the phase
shift curves of the individual amplifiers, as determined from Figs. 18
and 20, we are able to determine if the outer loop is stable with increasing
drive level. As an example, we shall consider a MAB circuit which has
a pole frequency at 5 kHz with a Q of 20. The design is such that

1

1

and

Vr,r 2 c 1 c 2 fl 3 d R 2 C 2

«.

= 2tt-5-10 3

In this case, the integrators will have approximately unity gain at the
resonant frequency of the pole. Consequently, all the three amplifiers
enter the nonlinear region at roughly the same level. From the given Q
we determine the phase lag needed to bring the main loop into oscillation
at the pole frequency. For the reasonably high Q of 20, this is given by:

<t>* = q = 0.05 rad S 2.9°.

Referring back to the phase characteristics of the inverter in Fig. 18,
we note that in the inverter, phase lag is introduced as the input level
is increased. However, this is counterbalanced by the phase lead intro-
duced by the integrator (see Fig. 20).

It is to be noted that for very high input levels where the phase lag
of the inverter becomes dominant, the circuit will, nevertheless, be
stable. This results from the sharp drop in the magnitude of the DF

2052

THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

when the amplifiers arc heavily saturated. The drop in the magnitude
of the DF has the effect of shifting the critical point toward the left
in the Nyquist plot of Fig. 21.

If the same circuit is scaled up to 27 kHz, the outer loop will un-
doubtedly be conditionally stable. In this case, both the inverter and
integrator provide phase lag (Figs. 18 and 20) and at the level which
produces 2.9-degree phase lag, the circuit will become unstable. As the
signal level increases, due to the instability, the circuit will reach a
stable limit cycle where the magnitude of the DF has decreased suffi-
ciently to just touch the Nyquist curve.

The method of analysis employed thus far has its greatest advantage
in that it is able to suggest many ways of getting around the stability
problem in MAB filters. These methods center in either improving the
manner in which the magnitude of the DF drops (possibly by making
this occur earlier so that it becomes dominant) or improving the phase
characteristics of the DF. One method which has been employed
successfully is the use of a diode network across the leaky integrator.
This has the effect of improving both of the DF characteristics, magni-
tude and phase. This circuit and the model for it are shown in Fig. 22.
The model in Fig. 22c is derived very simply by taking

i = i' and volt at op-amp input.

We then have,

id<eo>

DIODE
NETWORK

rr

9«irt

(a)

(b)

(c)

Fig. 22 — (a) Diode network across integrator, (b) I/O characteristics of the diode
network, (c) Model for (a).

STABILITY IN NETWORKS 2053

which if rearranged, we get the equation describing the model

6n =

[it - id{eo) ]

Drawing a phasor diagram for this circuit reveals that the magnitude
of the DF drops and simultaneously phase lead is introduced as signal
level increases. It is worthwhile to point out that limiting directly at
the output of the op-amp has the opposite effect (as shown earlier in
the example of Fig. 4); therefore, limiting cannot be blindly applied.
This technique has been employed in a MAB using 709's to obtain
a Q of oO at 100 kHz. Above this frequency, the gain of a 709 op-amp
is no longer sufficient to give a precision filter. Similarly, a MAB em-
ploying 702 op-amps was made to operate at 2 MHz with a Q of 10.
This method of nonlinear compensation has the disadvantage of
limiting the dynamic range of the filter. Therefore, techniques which
compensate the op-amps internally are more desirable. It has been
shown in Section II how internal compensation affects the DF of
closed loop amplifiers. In turn, these circuits may be rearranged to
produce better characteristics. We will not dwell on this subject in great
detail; however, we shall discuss the connection with slew rate. It is
well known that slew rato limiting in operational amplifiers is caused
by some form of either voltage or current limiting. Consequently, as
a result of this limiting, we may expect that the DF and, therefore,
conditional stability will be affected. However, this may not necessarily
be the case for it may happen that the limiting element causing slew
rate is not the dominant factor in determining the phase characteristic
of the op-amp. This is most vividly illustrated by a 709 op-amp with
input and output compensation. It has been experimentally observed
that the input compensation affects the frequency range of conditional
stability while the output compensation has a negligible effect. On the
other hand, slew rate limiting is mostly affected by the output compen-
sation. The dominance of the input compensation in controlling condi-
tional stability is consistent with our derivation of the DF of an open
loop 709 op-amp."

3.2.2 The Sinc/le Amplifier Biqnarl Filter

The single amplifier biquad' circuit is shown in Fig. 23a. In Figs. 23b
and c, we have the feedback structure, as seen by the op-amp, and its
Nyquist plot. Here, once again, we deal with a single op-amp in a closed
loop configuration and our interest lies in determining its stability.

2054 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

R 4

HWW

VW *-^AAr-^_

"c
AAAr

(a)

-ty\A< 1

c 2 c,

Ke

— o
e-E„

* \AAr-i

K=

R Q ll R c
Rb+b d ||Rc

Fig. 23— (a) SAB Circuit, (b) Feedback structure as seen by op-amp. (c) Nyquist,
plot of feedback structure assuming R 7 || Re » R*.

From the open loop DF curves of the op-amp, we can plot the locus
of the critical points as - 1/DF. As shown in Fig. 23c, the circuit is
conditionally stable as a result of more than 90 degrees of phase lag
in the amplifier. One manner in which stability can be achieved is by
appropriately including a diode network in the feedback structure such
as to move the circle in Fig. 23c to the right when the drive level exceeds
a predetermined threshold. This network can either be placed across
R 2 or R„ . Another alternative, which is advantageous from a dynamic-
range perspective, is to design the op-amp or its compensation such

STABILITY IX NETWORKS

2055

that proper phase lead is introduced sharply as the level is increased.
Modified forms of the compensation circuitry discussed in Section 3.1
can be used to achieve this effect.

3.3 Amplitude Stabilized Oscillators

Sine wave oscillators are usually designed in the linear domain by
placing the poles slightly to the right of the jw-axis in the S-plane. If
the network remained in its linear region, the network output would
increase exponentially without bound. Invariably, therefore, nonlineari-
ties are introduced to produce a stable limit cycle. Fig. 24 illustrates
a typical circuit which makes use of a saturation type of nonlincarity.
Since the DF of a saturating nonlinearity is real, the circuit will os-
cillate at the frequency of the tuned circuit, independent of the non-
linearity.

Many times, especially at high frequencies, nonlinearities with an
effective real DF are difficult to obtain. In these cases, the computation
of the DF (as a function of amplitude and frequency) is required to
determine both the amplitude and frequency of the limit cycle. To
illustrate this point, we refer to the oscillator of Fig. 25. (This circuit
is being currently employed in a 20-MHz subcable oscillator. In this
application, reliability necessitates an accurate determination of the
frequency of oscillation.) A first look at the circuit may lead us to the
conclusion that it behaves in the same manner as the oscillator shown
in Fig. 24a. The pair of diodes would behave as the saturating non-
linearity while the frequency selectivity would be provided by the
tank circuit. However, a second look reveals that this circuit is quite
similar to that shown in Fig. 11. It can also be seen, recalling Fig. 12,

— 1 >

+

Gfju))

-1

e out

=

1

-1

l m G(jo))

(o=0

(a) (b)

Fig. 24 — (a) Amplitude-stabilized oscillator, (b) Nyquist plot .

2056

THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

-c n : :

-VW

Fig. 25 — Amplitude-stabilized oscillator by the use of diodes.

that the pair of diodes is effectively in a feedback configuration with
frequency-sensitive elements, which produces a complex DF. Therefore,
we conclude that this circuit will have an oscillation frequency depen-
dent on the nonlinearity.

rv. CONCLUSION

A technique which analyzes the stability of frequency-selective
feedback structures containing nonlinearities has been presented. This
technique is an adaptation of the describing function technique to
include multiple feedback structures. This is accomplished by extending
the definition of the conventional describing function to include net-
works containing nonlinearities around frequency-dependent linear
elements. By doing so, the describing function of typical operational
amplifiers in open and closed loop configurations have been derived.
These, then, serve as a means of predicting the conditional stability
criteria in frequency-selective feedback networks such as the Multiple
Amplifier Biquad and the Single Amplifier Biquad. A valuable result
of this analysis is the discovery of circuits which circumvent in one way
or another the conditional stability of high-frequency biquads. This
result has had a highly beneficial impact on high-frequency biquads
since it relieves the conditional stability problem and shifts emphasis
to the maximum open-loop gain available at high frequencies. This
open-loop gain determines the precision which high Q circuits can
achieve at these high frequencies.

V. ACKNOWLEDGMENTS

Many of the techniques described in the memorandum came about
through the special efforts of R. L. Ukeiley. He has spent long hours

STABILITY IN NETWORKS 2057

in gathering important data relating to the maximum operating fre-
quencies for MAB's and open and closed loop op-amp measurements.
As a matter of fact, it was he who first observed the lack of correlation
between slew rate and conditional stability. I am also indebted to
J. J. Friend for his many suggestions and to R. C. Drechsler for his
patient efforts in guiding this project.

APPENDIX A

A Property of the Imaginary Part of A Describing Function

Theorem: The imaginary part of the describing function, G(A), associated
with a multiple valued input-output function, f(a), is given by

Im {G(A)\ =

-s A

ttA 2

where S A = area enclosed by f(a) for \a\ < A; positive when path taken
by nonlinearity is in a counterclockwise direction.

Proof:

1 C 2 '
Im [0(A)] = -j- / /(A sin o>0 cos cot d(cot) . (13)

At Jq
Let,

H — A sin cot.
Then,

dfj. = A cos cot d(cot).
It follows by substitution into eq. 13 that,

Im {G(A)\ =-h-6 /GO d» = -% Q.E.D.

APPENDIX B

709 Op-Amp Models

The schematic of the input circuitry of a j*709 op-amp is shown
in Fig. 26. The transistors Q 4 and Q 6 make up. the second stage of
amplification. Q 3 , Q a , and Q 7 decode the differential output of Q,
and Q 2 providing a single-ended signal at the base of Q 4 . The input
compensation, a series RC network, is placed across base and collector
of the Darlington pair, Q 4 and Q 6 . This provides (together with the

2058 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

_AA/\JL_ - INPUT FREQUENCY

COMPENSATION
V+

wv

Fig. 26 — Input circuitry of nA 709 op-amp. '

output compensation) the necessary frequency rolloff to stabilize the
amplifier in the linear domain.

Fig. 27 depicts the approximate circuit model of the second-stage
amplifier derived from the Ebers-Moll model." We have assumed that
the input impedance of the Darlington stage is much larger than R 2
and the load impedance presented by the emitter follower, Q 8 , is
negligible compared to R 3 . The cutoff region is represented by the
characteristics of V a , shown in Fig. 27b. Saturation is introduced in
the model by the diode D. In Fig. 27a the current / is given by

/ =

e in - V

R2 + Ri ~f"

Va - V*{V b .)
ft.

+ io(V )

Sd

Rewriting this equation, gives

V = ^e in +V a (V b ,)^-i D (V )Z 2

(14)

STABILITY IN NETWORKS

2059

where,

Zi — Ri -\- R 2 + -x-^-

Z 2 =

Z, +R 3

R, "

C, R,

H( — VW

T

>R 3

+ t :

+ 5^

v°) v »

v D

~JL

Fig. 27a — Circuit model for the second-stage amplifier.

v„*

SLOPE = _ M =_g R,

Fig. 27b — Nonlinear elements of the second stage amplifier corresponding to the
cutoff region.

Vcat * 0.2

Fig. 27c — Nonlinear elements of the second stage amplifier corresponding to the
saturation region.

2060

THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

We also obtain from Fig. 27a

R 2

R2 1 T r ^2

(15)

where,

Z a =

ZiR 2

Z± — R?

Figure 28 is a block-diagram representation of eqs. 14 and 15. For
simplicity, we distinguish between the cutoff and saturation regions.
The result of this is shown in Fig. 29. In both models, the forward path
containing Z 2 /Z v in Fig. 28 is neglected since the other path through
V a has a much larger gain. We have also assumed, with some loss of
generality, that the biasing is such that a sufficiently high drive level
producing cutoff (saturation) does not cause saturation (cutoff) during
some other portion of the cycle. However, in general, the problem can
be solved by computing the DF for the inner loops involving i D and V a
and employing these results to solve for the DF for the complete
system.

The model of Fig. 29a represents the second-stage amplifier when
driven into cutoff. It is obtained directly from Fig. 28 by making a
slight rearrangement at the output in addition to the two previously

/ SLOPE=-n

5 V be ( V DD V a .[~Z7
— v-^

Fig. 28 — Block diagram representation of eqs. 14 and 15.

STABILITY IN NETWORKS

2061

-fj, [SC,R,+I]
[SC,R 2 (m+1)+1]

(a)

\^f

V a

—

2 2 R 2

Z .
R 2

\v be

Z 1 R 3

J

j D

V o

R 3 2 1

R 2 m

j D

(

(b)

Fig. 29 — (a) Model for the cutoff region, (b) Model for the saturation region.

mentioned assumptions. The network parameters are given by

Z0R1

SCfii

zlR 3

SCM + R 2 + R 3 ) + 1

/to

SC.fl, + 1

z.

SCy(R X + B a ) + 1

z>

SC 1 (R 1 + fl 2 ) + 1

Rn

SCi/^2

The model shown in Fig. 29a depicts the saturation region. It is obtained
from Fig. 28 by restricting V a vs V be to operate in the linear region.
Consequently, this nonlinear element (V a vs V bt ) in Fig. 28 becomes a
linear gain element. The process of reduction is shown in Fig. 30. Figure
30c, after factoring out the feed-forward path and assuming a large /x,
yields the saturation model of J 1 "ig. 29b. The linear element in the feed-
back loop in terms of fundamental parameters is:

R:\Z X

R2H

Ik.

R 2 n

ggigg; + gg) +

SC\

"1

2062 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1972

(b)

/~\-

~P

R 2

2 2

v

R 3 Z 3

1

z 3

R 3 Z 3

M« 2

^

(c)

H/xR^/RjZ,

M« 2

r

Fig. 30— Reduction of Fig. 28.

REFERENCES

1. Friend, J. J., "A Single Operational Amplifier Biquadratic: Filter Section,"
11)70 IEEE International Symposium on Circuit Theory Digest, December
14-16, 1970, pp. 179-180.

LiOICliaiD, Jj. l_/., WIl florae llUIUUieui i neuuiiiciia in iieguiiiiui^ uj-o^.»»«o,

Automatika i Telemekanika, 8, No. 5 (September-October 1947), pp. 349-383.

4. Gibson, J. E., Nonlinear Automatic Control, New York: McGraw-Hill, 1963,

pp. 203-235, 343-438.

5. Siljak, D., Nonlinear Systems, New York: John Wiley, 1969, Chapters 3, 4, 5, 6, 7

and App. F and (!.

STABILITY IN NETWORKS 2063

6. Krylov, N., and Bogoliubov, N., New Methods in Nonlinear Mechanics, Moscow:

State Publishing House, 1934.

7. West, J. C., Douce, J. L., and Livesly, R. K., "The Dual Input Describing

Function and Its Use in the Analysis of Nonlinear Feedback Systems."
Proc. IEEE, 108, Part B, 1956, pp. 463-474.

8. Op. Cit., Siljak, D., pp. 472, 481, 498-510.

9. Ukeiley, R. L., private communication relating to unpublished data concerning

frequency and slew rate limitations.

10. Giles, J. N., Fairchild Semiconductor IC Applications Handbook, Fairchild Semi-

conductor, 1967, pp. 57-59.

11. Angelo, Jr., E. J., Electronics: BJT's FET's and Microcircuits, New York:

McGraw-Hill, 1969, pp. 245-250.